Siegel's lemma

In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue;^[1] Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929.^[2] It is a pure existence theorem for a system of linear equations.

Siegel's lemma has been refined in recent years to produce sharper bounds on the estimates given by the lemma.^[3]

^ Thue, Axel (1909). "Über Annäherungswerte algebraischer Zahlen". J. Reine Angew. Math. 1909 (135): 284–305. doi:10.1515/crll.1909.135.284. S2CID 125903243.
^ Siegel, Carl Ludwig (1929). "Über einige Anwendungen diophantischer Approximationen". Abh. Preuss. Akad. Wiss. Phys. Math. Kl.: 41–69., reprinted in Gesammelte Abhandlungen, volume 1; the lemma is stated on page 213
^ Bombieri, E.; Mueller, J. (1983). "On effective measures of irrationality for ${\scriptscriptstyle {\sqrt[{r}]{a/b}}}$ and related numbers". Journal für die reine und angewandte Mathematik. 342: 173–196.